.

REPORT’ No. 251

APEROXHVIATIONS FOR COLUMN EFFECT IN

AIRPLANE WING SPARS

By EDWARD P. WARNER and MAC SHORT

Massachusetts Institute of Technology

https://ntrs.nasa.gov/search.jsp?R=19930091318 2018-06-30T17:44:13+00:00Z

REPORT No. 251

AI?PRO.XIMATIONS FOR COLUMN EFFECT’ IN AIRPL.AM3 WING SPARS

By EDWARD P. WARNER and MAC SHORT

SUMMARY

The significance atibaching to” “column effect” in airpIane wing spars has been increasinglyrealized with the passage of time, but exact computations of the corrections to bending momentcur-w resulting from the e.xktence of end loads are frequeMIy omitted because of the additionalIabor imvolred in &n analysis by rigorously correct methods. The present report, preparedfor publication by the National Advisory Committee for Aeronautics, represents an attemptto provide for approximate column effect corrections that can be graphically or otherwiseexpressed so as to be applied with a minimum of Iabor. (km-es are pIotted giving approti-

ate values of the correction factors for singIe and two bay trusses of -wrying proportionsand with various relationships between axial and lateral loads. It is further shown from ananaIysis of those cum-es that rough but usefuI approximations can be obtained from Perry’sformuIa for corrected bending moment, with the assumed distance between points of infectionarbitrarily modified in accordance with ruIes given in the report.

The discussion of general rules of -rariation of bending stress with axial Ioad is accompaniedby a study of the best distribution tif the points of support along a spar for ~arious conditionsof Ioading.

GENERAL N’.4TLTREOF PROBLEAf

The -ritaI significance of coIumn effect in a sIender beam cam-y-kg both lateral load andcompression has long been realized, and formuke for deaIiug with the case of the IateralIy loadedstrut and calculating the equi-raIent bending moment at various points of such a member ba-rebeen a-raiIable for many decades. In cid and mechanical engineering structures, how-e-rer,the occurrence of beams stressed in flexure and carrying also a compressi~e load approximatingto the “Euler load, ” or that which vvouId cause fa.i.kre by Iateral instability -ivith lateral Ioadentire~y lacking, is rare. Only with the coming of the airplane did the use of members workingunder such conditions become a commonplace of design, and only then did the need for accurateand complete means of calculating them become acute.

The first efforts to treat “column effect, ” or the motivation of the bending momentdiagram b-y the amount of compression moments dependenk on deflection xere made throughapproximations, of which a great number ha~e been devised. One or more are gi-ren in e-rerywork on the strength of materials, and &Tem-elllists @eference 1) eight approximate formuke,dl intended to serve the same purpose and giving widely -varying results in some cases. N’otuntiI 1916 did a complete and rigorously accurate method of calculation of continuous beamsunder combined load become a-vaiIable. (Reference 2.)

With the development of the “Berry Method” or “Exact Method” or “GeneralizedTheorem of Three Moments, ” as it is varioudy called, the problem of column effect passesinto a new stage= Accurate calculations could be made, subject to the usual assumptions abouthomogeneity of material and the absence of &continuities of section, and the questio~ thenbecame one of whether or not it was worth whiIe to go through with the somewhat eIaborate

45

486 REPORT NATION.4L ADVISORY COMMITTEE FOR AERONAUTICS

processes involved in any particular case. In many instances it is not worth while, or is notconsidered by the designer to be so, and although the generalized equation has had a steadilyincreasing use in practice its employment is still far from universal.

Taking W this into account, the desirability of having some reliable means of approxim~tingthe column effect in advance is obvious. Such a trustworthy means, if one can be secured, willshow in some cases that column effect is unimportant and that no check ca]cuIations includingallowance for it are necessary, while in those instances where it is of importance the inclusionof a preliminary allowance for its magnitude will decrease the liability of having to make twoor more tentative spar designs with a calculation for each one.

The difficulty of making exact calculations is noi onIy inherent in the form of the equations,which require for their solution perhaps triple the time needed for the analysis of a. continuousbeam by ordinary methods, especiaNy in view of the fact that some of the resuIts from thegeneralized form corm out as the relatively small difference of two very large quantities and thenumber of significant figures that have to be preserved ~hrough the preliminary operations istherefore greater than would be necessary for a corresponding degree of accuracy if the equationwere left in its familiar form. The labor of a solution is further increased by the fact that thegeneralized method is only a check, which can not be applied until a preliminary design of thvspar section has been made. It is necessary to work”the problem through once in order to arriveat that preliminary design and then to work it through again to be sure that, Lhe preliminmyselection would fit the requirements. Furthermore, in case the check shows the sect io[~ firs tchosen to be inadequate or overstrong still another complete set of calculations must be madeafter the redesign.

The stresses in an airplane wing sptw depend on certain properties of the spar section, thew-ing truss form, and the loading, and for any given general form of truss the numbm of suchproperties exerting an influence is definite and small. In a two-bay truss with the spars con-tinuous over three points of support and hinged at their inner ends, for example, the bentlingand direct stresses depend on the leng%h of each of the three bays of the spar, on the momentof inertia, section moduIus, and area of its cross-section, and on the lateral load per unit of Icngthand on the compressive or tensile load applied. The compressive lo~d, however, is itself depend-ent only on the laieral load, the distribution of supports along the spar, and the gap (if it bea biplane that, is under consideration), and the variation of stress with change of linear dinlen-sions in a truss in which geometric similarity is preserved, foHows simple and well understoodlaws. The number of vari~bles can therefore be materially reduced if all factors involvinglinear dimensions are converted to a constznt overall length of spar as a basis of comparimn,and coefficients of bending moment figured without regard to column effect can be plotted, forexample, in terms of the percentage of tota~ length of spar in the inner bay and the percentagein the overhang as the only two independent variables, Since the inner bay, outer bay, andoverhang sum to 100 per cent, any other pair from among those three might be selected insteadof the inner bay and overhang as the basic quantities governing the magnitude of the bendingmoment coefficients.

A compIete soIution without column effect has been made and published for two-baywing trusses of proportions extending over the full range of probable design practice. (Refer-ence 3.) It is only a little more difficult to do the same thing with column effect taken intoaccount, and the work has been carried through for two typicaI wing truss forms, the singlebay with overhang and the two-bay truss treated in Technical Report No. 214 (Reference 3),the bencling moment at the inner end of the spar being assumed zero in all cases.

SINGLE-BAY TRUSS

MThcn a spar is continuous over only two supports and is pinned at the inner end, the bendingmoments at the support are obviousIy independent of any column action, that at the inner endbeing zero while that at the strut is governed only by the length and loading of the overhangingportion of the spar. The distribution of moment in the bay, however, is directly affected by

APPROXIMATIONS FOR COLUMN EFFECT IN WING SPARS 487

the compression coIumn moments of magnitude Py, where y is the deflection of the spar afteralI loads are on it, having to be added to or subtracted from the moments due to Iateral Ioad,the si=n of the correction of course depending on the siams of the lateral load moment and of thedeflection. In general, it is expected that the correction to be applied to the maximum -dueof the moment iri the middle of a bay wiU be positive, as a spar running o~er ordy two supportscommonly deflects in the direction of the lateraI load, and that gives an additive cohm.u correc-tion. It will, in fact, always do so unless the ratio of length of overhang to length of bay isexceptionally great.

The formuIa for the maximum -m.Iue of a bending moment attained between supports isgiven by Berry as:

where r is the distance frcm the ~oint of maximum moment to the middle point of the ha-r, andis in turn defied by: *

and the other symbok are:P= compressive loadE= modulus of elastici~y of material1= moment of inertia of cross sectionw = laterai load per unit length of spar

d-

P‘= H1= length of bay of spar

7

.

M. and MB= moments at ends of bay

4-

ZPa=ap=~ EI”

In the particular case under consideration MB is O, and the form becomes somewhat simpli-fied, beiug most conveniently expressible for most calculations in the shape:

itwilI be obserred in this formuIa thati each term varies as Wlz if CKis Icep&constant. It is pos-sible to consider ordinary bending moment stress charts, such as were given in Report LNO.214(Reference 3), therefore as representing the speciaI case in ~hich a equals 0° (no compressi~e .load), and simiIar sets of charts could be developed for any other ,value of c. For a given dueof that, quantity, in other-words, coefficients of bending moment can always be written independ-ently of the absoIute -values of the lateral Ioading, the dimensions of the truss, and the dimensionsof t-he spar section, -rarying only with the distribution of the struts aIong the spar, or in thisparticular case with the placing of the single strut.

The calculation has been made for fi~e cases, with the length of the effective overhang22.2, 30, 35, 37.5, and 40 per cent of the total effective overall Iength, the true length havingbeen reduced for the tip Ioss aUowance. The beding moment at the strut, fl results beingreduced to a total spar length of 100 inches and to a loading of 1 pound per inch of Iength, wouIdthen be 246,450, 612, 703, and 800 pounds inches in the five cases, while the makimum bendingmoment in the bay tith no coIumn effecti would on similar assumptions be 638, 408266, 200,and 139 pounds inches.

$24s.+27—32

488 REPORT NATIONAL ADVISORY COMMITTEE FOR AERONAUTS

Making similar calculations of moment in the bay for the five cases wiih various values 0[c ranging from 15° to 85°, the vaIues found to change are tabulated below and plotted in Figure 1.

The statement of a as an angle is of course a mathematical fiction. It is actually a purenumber, which becomes equal to ~/2 when the point of coIIapse as a column is reached, but

.d

Fm. I

since the development of the theory brings in certain terms which can be best expressed bytrigonometric functions, it is sometimes convenient to state the variable as an angle. The

1d-F— ~of course gives the angle in radians.

‘beet ‘O1utiOn’ a=% El

I 1“. I

1Moments In bay

I

Momemt in bayMoment without column effect

I.— 1.:Per cent length of OTerhang

I

22.2 Ii, 30 , 35 \ 37.5 40

la”2130135137”51401,— ~—.— :— —— —— f— ~—

o“ 638 408 ; 266[ M)! 13915”

I. CO/ ;:$ I.fll653 416 270 ‘... ------

I.LMI l.m I140 I.02 ..-..i-%. 1.01

30 710 446 288; 211 141. ;:=;; ~ ~. Og

1.0845 843 :; ;. . --------

1.01530 142

601.32; 1.30

1,124f 680L 22 . ..- . . ..- 1.02 !

261 139 1.761 - 1.49 \ 1,31 1.04 ‘‘ ::;’

:: . ..-?.!?- ;;% ~ 8S0623 -----% ;-] l+; ~:::::;:::l 485 . . . . . . . . . . . 3,264 ~ 1,536

I 1] {g] $; [---”~iti~:% I1

, .3.36 .0-4 ;,, !_

- -.--— r

As the computations in making up this tabIe were never carried beyond four s@ificant.figures, the last digit in the moment figures is somewhat uncertain but the probable error isnot in excess of 1$.

In Figure 1 the moments are plotted directIy, while Figure 2 shows the ratios of cor-rection factor required by column effect. It will be observed th~t there is comparativelylittle difference among the first four curves in Figure 2, suggesting the possibility of using astandard curve. of the correction factor to be ap~lied to moments in the middle of the bay,and the value of a for all single bay wing trtisses with overhang lengths of Iess than 37 per cent

—

,—

APPROXIMATIONS FOR COLUMA- EFFECT IA-WING SPARS 489

provided a does not exceed about 45°. I@re 3 is pIotted with the same ordinates as Figure 2,~EI

but with the abscissa changed to the ratio ~, where P, is the EuIer Ioad, ~. This ratio is

()

2 c

equal to ~~ -

In Figure 1, on the curves corresponfig to the 30, 35, and 37.5 per cent, overhangs thevalue of the moment ah fihe strut has been indicated by a shork cross line. Even with an over-hang as Iong as 37.5 per cent, approaching a semicantile-rer form, the moment in the bay super-sedes that at the strut as the criticaI element in the calculation when the end Ioad is more than90 per cent of that under which the spar wouId coIIapse as a simple coIumn pin-jointed at its ends.

While it would be possibIe, as just noted, to pIot a sing~e curve of moment correction factorsfor singIe-bay trusses which wotid hold within a maximum error of 10 per cent or 15 per centfor EN cases likely to be met with in practice, except those with overhangs so long that theywould ordinarily be described and thought of as semicantiIever types, it would be better eitherto continue the use of se-veraI such curves as those in Figure 3 and to interpolate for the par-

111111 [11[

11111111111

.

..

aFm.2

ticdar Iength of overhang under consideration, or, alternatively, to find some general approxim-ate method, either emperical or rationaI, which comes sticiently close to fitting aII cases.

An approximation much used in such cases is the Perry formula. (Reference 4.)

~,_xx P.P.–P

where P is the end load actually carried, P. the EuIer load, 3.2 Lhe moment under lateral loadalone, and M’ the moment with alowance for column effect. Since the formula wasdevised for application to struts carrying a lateraI Ioad and pin-jointed at their ends, it ismanifestly unfair to aUow nothing for stiffening of the spar by conbuity past one or both ofthe ends of the bay, and in actual use the factor P= is commonIy replaced by P;, equal to-277T

‘YY 1t is the distance between the points of inflection rather than that between the ends

of the bays. That distance itself, of course, -raries with the length of overhang, a curve of ~

490 REPORT NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

for the single-bay spar with hinged inner end being plotted against perce~tage length of overhangin Figure 4.

Unfortunately, the points of inflection do not remain fixed, but move outward toward thoends of the bay as the column effect becomes increasingly important. The writer has made it

a practice to allow for this shift empirically by basing the Euler Ioad in the Perry formula neit,he ron Z nor l’, but on a fictitious length 1’ + n(l – l’). The coef%cient n is, of course, fractionaland varies with the Lype of truss under considers tion. For the single-bay truss with the sparpin-jointed at the inner end experience shows that a value of o.8 for n gives best results.

. 100

80

60

lfz

40.

20

0 10 20 30 40 507. Overhang

FIG, 4

The Perry formula correction factors have been calculated on that, basis, and the rxtio ofthe correct maximum bending moment in the bay to that approximately determined hy theformula is plotted in Figure 5. The 40 per cent overhang has been omitted in both cases, asit is manifest that no correction factor is needed in that case, and any facior or method whichincreases the value of the moment will be worse than none at all.

APPROXIM.4TIONS FOR COLUMN EFFECT IN WING SPARS 491

It is evident that the Perry fornmki gives excellertt results in general. The dues for themaximum moment derived by its use are correct within 5 per cent for all m.lues of a up to68° with a 22 per cent o-rerhang, 76° with 30 per cent, 44° with 35 per cent, and 32° with 37.5per cent. Most of the cases met with in practice wiU show values of a Iower than these, andit can therefore be said in general that the Perry formuIa gives approximations sufhientlyaccurate for ordinary use nit.bout any exact c.aIculation except when the overhang is unusuaIIy

‘i,,:l;~‘-‘:,~;‘-4 Ill II II Ill!k,J Ill II I II 1 IQ.u

$+ Illtll “Illlt IL

.s ~ ,.~“111[1 Itllll 30 %~

‘

& 11111111$?E ~[. [ 1111 1111 I@ It I I I I I 1 +2$++

;- ~.$$

1111111~1 “1 I LM ‘1g $ /.0Es

1~ ~!l i I i I 3+~L1~o

111- T741vl

i i“’g I“ltllm r%, r--l I

$08 1111 1111 1111

“ I 1111111111 \lll

~7 111111111113 7.:<%I

Iill 11111 IT Io“ /00 20” 300 400 50° 60” 70° 80°

aFLG. 5

Long and the compression in the bay large. Exact figures can be read off, of course, by inter-polation between the curves in Fi=gu-e 3 if t.hatiis desired.

The sudden change of form of the curve of mria.t.ion of maximum bending moments withcompression in the bay in going from a 37.5 to 40 per cent overhang is startling at first sight.It is, howe~er, entireb logical, and it would be expected that there would be a ~ery sharp, ifnot absolutely discontinuous, passage from beams in which the moment increases very r~pidly

;QO 80 L70 40 20 bz span

FIG.6

at large ~aIues of c to those in which it decreases with corresponding rapidity in the sameregion. The reason is thab when CYapproaches very closely to 90° the compression momentsbecome the controlling factor, and whether those moments act with or against the effect oflateral Ioad in the middle of the bay depends on the form of the initial deflection curve, which,in turn, is governed by the length of o-ierhang. Figme 6 illustrates diagrammatically thegeneral form of deflection curves that would be found with 30 and 40 per cent overha~g, respec-ti-seIy, and the variation of the curve with changing a in the two cases. Manifestly, when a is

492 REPORT NATION.4L ADVISORY COMMITTEE FOR AERONAUTICS

very Iarge it is almost certain that either the part of the beam which was iniiiaI1y deflectediibove the datum line or that which was deflected below wiH have taken control, and the columnmoments will have thrown all the deflections in one direction. Such a condition as is indicatedin the third diagram in Figure 6 is conceivable, but most unstable and unlikely.

In order to illustrate the amount of work involved in an accurate calculation for one over-hang the compIete figures for the case of 40 per cent overhang are tabulated below:

ILLUSTRATIVE CALCULATION

Length of bay=tW’Length of overhang= 40”w =1 Ib./i~. W =3Q0 lb. in.

1-a (deg. )------------------------------------------------a (rad. ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0. 261&sOca . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..- L 03528mca . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..- 3.36371

~.= (b) ‘-------------------------------------------.008727

I

P--------- "-"----"-----"------------------------------ 13130.2

MA me a --------------------------------------------

1

3090.97”M&s6Ca.............................................._ 826.22

~3-~)s,ca=(A)--. ..................... BW3.2

o.523%1.154702. m

.017453

3232.92

16C0.03923.76

3328.90

‘= MA C-ic’a—=(tanw)---------------------------

“’’(5-%9.11724 .2403

/IZ(deg. )-. --.. --... -..-----. --.. -.-. ---. _---__----_q 6“ 41J L3:~3:;@ (rad, ). .-.. -.--. -.--.. -.-—_-- .._. -_--. -------. -----+ .1167 ;$.--_ . ..-.. -.-... _.--_ --.. _-_-. -.---. ---_ ------------i 13.37sec W-------------------------------------------- ~ ~CJ69 : ;:&J?:see m see a-- . . . . . . . ..--. -i—-. -------—-. -—----—--- . 24, .

Mwor.=~(A) (see Kz) -p) . . . . . . . . . . ______ 139.7 I 140.9

—

::7:1:41421

.026180

1459.02

1131.371131.37

1497.6$

.3777

20”41’.361013.79L0689L511714L8

643L 0’47202.m1.15470

.034907

823.68

923.7616C0.00

841.36

.5489

28” 46’.502014,38

1.14072.2814

139.0

I75 ; 801.3m ~ 1.396273.86371~ 5.753771.03528, 1.01543.OWA2 : .646542

b.m5. 25

I4EI.05

828.22: 812.3436’30.97 4607.02

483.95 ; 355.01

.3556 / 1.1442I

40° 33’ : 48° 51{.7078 . 85%16.22 ! 18.32

1.3160 : 1,5190.......................

llL. 6 : 77.8

1

.040451 !

803.00;9179.Oa ,

102.46 I

3.919[

75” 41’ ‘LJy I

4, ok6 ;

1.. . . . . . . . -.’

5.5t1 -.

LOCATION OF STRUT

The most economical strut location in a single-bay truss with untaperecl wing spars, con-siderirw ordY spar weight, is obviously that which makes the moment at the strut equal to the-..maximum in the bay. The percentage length of overhang to be used to conform with thatspecification is plotted against a in Figure 7, and it will be observed that the variation in strutlocation with change of a is very slow. To determige the exact value for any particular casea should be found in terms of other and more generaI properties of the airplane.

40~ .$L35~‘a

830

0 20” 40“ 60” 80”C%

FIG. 7

In the particular, and v-cry common, case of the biplane with upper and lower wings ofequaI size, and with the Ioadj.ng onupper, the compression in the upper

the lower wing assumed to be 90 per cent of that on thespar is independent of strut location, and is always:

~=L9 wl’_jlgG.f2G .

where P is the compression, w the load on the spar in pounds per unit of length, 1 the totaleffective length of the spar (with the correction for tip loss taken off), and G the gap. Themoment of inertia of the spar by the familiar formula for bending stress is equal to:

APPBOXIMATIOXS FOR CIOLJIZWITEFFECT! IN WING SPARS 493

c being the distance from the neutral atis to the outermost fiber. On the assumption that thebending moment i.n”,the bay is to be equaI to that at the strut and that the latter can thereforebe taken as one of the worst comlitiom, I becomes for & symmetrical spar:

Wlizoz4fb

/0 being the Iength of the o~erhang and d the depth of the spar. Then, 10beiug the length ofthe bay,

P ~ ‘Y3.8j,[)–~1= .L, ‘GdE

7 p. M f, -“ML~=&.2> E’I ~ l.’ x 1.05GdE–> 10E 1.05Gd

fb can be determined in terms of the ultimate totaI stress.

Xc

On the assumption, pretiousIy discussed ekewhere (Reference 5) that the radius of gyra-tion of an airpIane wing spar section is 0.36 of the depth,

to a range of fi of from 0.61 to 0.78. fb

J*Either in spruce or iu duritumin, ~ will then be between

~& and &o, with the smalIer values going with a short overhang.

Taking the factors entetig into the expression for a separately, # may vary 1.5 to 3.0, $0

from 2.2 to 4.6, md ~ from about 2.5 to 53. For auy particular values of those quantities

cr can be calculated from the formda, and the due so fouud should be equaI to that whichmakes the moments at the strut and in the bay equal for the o-rerhang Iength uuder considera-tion.

The method of determhxin g optimum overhang can best be demonstrated by a particular

probIem. ~ -wi.l.Ibe taken as 3.5, ~ as 40, and a series of o-rerhang lengths tried.

.——

.—

494 REPORT N.4TIONAL ADVISORY COMMITTEE FOR AERONAUTICS

These points have been plotted in the dotted line in Figure 7, and the intersection of thetwo curves defines the best effe.c~ive overhang at just under 35 per cent (equivalent, w;th theusuaI deduction of one-fifth or one-sixth of a chord length from the end to allow for tip losses,to about 40 per cent true overhang).

The solution of other problems of a similar nature gives the data for Figure 8, where curves7 7

are plotted to show the best length of overhang in terms of ~ and -~. The extreme range of

useful values, it wiH be noted, is from 31 to 39 per cent.

5

4

Z/G

3

220 30 40 50 60z/d

FIG. 8

TWO-BAY TRUSSES

The same general mekhod can be used for investigating the magnitude of column effeebin wing spars continuous over three supports, alihough of course the number of variables steadilyincreases with increasing number of bays. Such a study has been made for a particular caseof the two-bay spar pin-jointed at its inner end.

The trusses selected for calculation in this case have the proportions listed be~ow:

IKo.

Length OuterI

t inner bay bayOverhang

IPercent Percent Per cent

1.............. 32.5 45 22.5 ;~ 11. . . . . . . . . . . . 29.5 54.6 1:.9

111. . . . . . . . . . . 33.3 3s 5 28,2IV . . . . . . . . . . . 37.1 , 429 20

I v 27.1 50 229. . . . . . . . . . . .i VI . . . . . . . . . . . 45

I \’II.......... :! ~ 45 ;;YIII . . . . . . . . . 30 , 45 25lx.. . . . . . . . . . 40 50x . . . . . . . . . . . . 31.7 ~ 4’s.3 il ~

I

The calculations for the first case, and for the first alone, were carried through completelyfor seven different values of a ranging from 15° to 85°. The general form of the curve of varia-tion of moments with changing compression in the spar having thus been determined, only afew values were seIected for each of the other cases. The inner and outer bays of course havedifferent values of CY. The larger of the two vaIues was in every case used as a key number forindexing. In other words, when a is mentioned in a tabulation or pIot it is, unless otherwisespecified, taken from the bay having the largest a. That is true even when the specific quantitytzbulated or plotted in terms of a relates directly to the other bay, the one with the smaller a.In most two-bay spars a is la~ger in the inner bay than in the outer, and it is therefore in theinner bay that the column effect is most critical, but if for any reason the ratio of length ofouter bay to inner is made especially great that condition is changed, and it is between the innerand outer struts that the spar comes nearest to fai~ure as a simple coIumn. Among the cases

here discussed, a was greater in the outer bay than in the inner only in NOS. II, Y, and VIII.A beam continuous over three supports and pinned at the inner end of course has a zero

bending moment at that point,, and the moment at the other end support, corresponding to theouter strut, is dependent only upon the properties of the overhanging portion and not on the

‘

APPROXIMATIONS FOR COLUMN EFFECT ~“ WING SPARS 495

compression in the bays. The moment at the middle support, however, ~aries with the com-pression—a fact which is not ta.ken into account by any of the ordinary approximations forcolumn action. It is manifest thab in general the bending moment at the intermediate supportof such a beam will be increased by compression in the bays, as the general direction of the deflec-tion is the same as that of the lateral load, and the application of compressive force tends there-fore LOspring the mid-point of the beam up a-way from its resiraint at the support with increasedfame. The reaction at the middle support is therefore increased and the bending moment mustincrease in absolute value likewise.

The way in which the magnitude of any increase depends on a and on the distribution ofthe supports of the spar is best shown by tabulation and pIotting of pdicular exampIes. ThevaIues of MB (B being the middle support) for the nine cases previously described are tabuIatedbe~ovr, -ivhiIe curves of MB against a are plotted for five of them in Figure 9.

350 II IHIII III L?

II 11111111 /

~11,1111~=,m Ill I\ Ill

~~~ ill. lll~ ‘illltt

iii’ iltlll 1111 /,~ Illlltllll II

X3 [111111]IIIIH

[50 ,,1- ir1111111”!

—,, ,,, = ,,

~ltll I

5“ iiii~i i[iiil~ii~

1

,,. ,,, , ,

Jo I toI JoeI Joo I I Mllllll40” 500 70” 80“

clFIG. 9

Figme 9 hatig shown the variation of the absoIute ~alues of the momen~, Mgurc 10represents reIati~e variation for ratio of the moments of middle support. for various vaIues ofa to the corresponding figure with a equal to zero (column effect absent). The cur~es areplotted against CYand each one represents one of the typica~ cases calculated.

To correlate these results and make it possibIe to predict probable -ralues of Jf~ as far asthe proportions are diEerent from these here tried, a contour chart. was constructed wifih curvesrepresenting equal -dues of the relative eoIumn etlecfi or percentage correction to X~ ah agiven a. The form of the chart showed that the correction fac~ors -were substantially a factor

———

—

.—-—

.—

Corr.ecfionfacfor for cO/umn effecf on M.

I

I

APPROXIMATIONS FOE COLUMN EFFECT IN WIL”G SPARS 497

the exact method for the particular dimensions in question. Secondly, it greatly reduces thetime that sometimes has to be given to repeated triaIs to determine the correct section andcorrect dues of a for the outer and inner bays. Manifestly any change of MB also changesthe compression in the outer bay a~d the -m.Iue of a in that bay. That makes astonishinglyIittIe difference so long as a in the outer bay issmaIIer than that in the inner bay. A arbitrary

reduction of a in the outer bay in case 1 by one-third chaD~ed the bertdinga~ the middIe support

by lessthan 2 per cent, so Iong as a did not exceed 82° in the inner bay. In the general casein vrhich the inner bay is the critical one, recalculation to allow for change in compressionin the outer bay is therefore unnecessary, buh when the outer bay is the larger vaIue of a thosechanges become very serious and anything that can be done to facilitate their prediction hadvance and without any repeated trials, as these charts faclitate it, teds toward increasedaccuracy and economy of time.

MOMENTS WiTHIN THE BAYS

The maximum values of the moments within the inner and outer bays, as weII as those ofMB, the moment at the inner strut, ha-re been calculated directly for a number of ilIust.rati~ecases, inc~uding aIl except ATOS.lT1, VIII, and IX of the 10 already tabulated and any addit-ional one, NTo.XI, introduced to cover the case in which the values of c in the inner and outerbay are very slightly unequal.

AS a first step in the systematic treatment of the results of these cakulations 60° hasbeen selected as a standard value of a and the figures for the seven strut distributions triedare tabuIated herewith for that value. As before, the bay in which a is the largest, whetherinner or outer, is aIways taken as the basis of rating, and the value of a is considered as fixedat 60° in that bay.

1 iI Lengths Bay hatig18rg-, cOrre-

Case ~ & mm ofa qmding I~w~ tit bay‘.~~;her~

Innerbm OuterbAY~Overhang

t1-------.---;

..-j..::J___=’ %! :!1n---.-.-.-.-,------:-!---.-----_----!--:---.-----outer------------;

; ~l-.--------L.--._-------~--.-----_/______ Inner-——__~

h v------------Ion::-=:::::::::~~~ :

~ vm--.---.-.-”-----iwp-----i;:x:lT------x:lT-iher------------fI x.-..-------- 3L7 ; 43.3 ,

xl----------- 32.5 i 47.5 I ~g I~$~----.--- %: ~t 1

------------- 57.L ~,

I1

! M-==. inI

Case j b?y where]alslargest

It i_

i =L--------lI

—-------- .! &.___J

I

------- ----v_____-l

vIL______

4&?&::~ xi_. ___-,

M.a. inth&:h:3

columntx%ct

7.5191119104113S556

1U363

I-AI! 66L43 {L34 –3:L4S ; 45~~ 26L 40 79L 10 -.-------=-L53 . . . . . . . . ..-L31 129

-1

-!~___2:.7s :

33z I lx ‘

.—

.———

——

I I------------------------------- ,

91‘---i--------------------!LAQ

The fact most strikingly evidenced by the table is that column effect is always of negligibleimportance or actuaHy negative in the bay, having the smalIer a, unless the kwo bays are ~erycIose together in &hat respect. The correction factor for the momerit in that bay exceedsunity only when the inner strut is within 4 per cent of the length of the spar from the positionwhich wouId make the two vaIues of a exactI-y equaI.

498 REPORT NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

The explanation is of course essentially the same that served to ac~ount, for the falling offof mid-bay moments with increasing compression in a single-bay spar with a long overhang.The deflection being largest in that bay in which a is largest, that portion of the spar “takescontrol” as the compression is increased and forces a steady decrease of the deflection in theother bay. An actual negativing of the column effect and decrease of mid-bay moments inthe bay with the smaHer a resuIt, in many cases. The nature of the effect is graphically shownin Figure 12, where the deflection curves for the spar covered by Case 1 are plotted for threevalues of a (in the inner bay). These curves, it should be noted, are not diagrammatic sketcheslike those of Figure 6, but are the resuIt of actual deflection calculations and accurately repre-sent the change of the form of the elastic curve for a particular spar as the compression varies,the lateral Ioad remaining constant.

The comparatively slight change of the deflection curve with changing a when that quantityis below 60° and the much more rapid change as 90° is approached is strikingly shown, as is themanner in which the inner bay ‘(takes control ~’ when the compressiort is very large so that thespar is finally forced down below the Iine of supports everywhere in the outer bay, giving a nega-tive column effect on the moments there, even though the line of the spar continues to showa double reversal of curvature within the bay. Even under the largest end loads the deflectionsin the bays are small compared wi th that at the tip of the overhang, which to be sure is unusualIylong in this particular spar.

FIG.:2

In the bay whero a is Iargest, and where co~umn effects are therefore uniformly positive, thecorrection factors on mid-bay moments range from 1.10 to 1.54 in the group of illustrativeproblems considered. There me hardIy enough points to plot a satisfactory contour chart ofcorrection factors in terms of the dimensions of the truss, and therefore the figures can mosteasily be systematized by reIating them to the factors given for corresponding conditions byPerry’s approximate formula, exactly as was clone in examining the moments in the singIe-bayspar.

Again the value of n, the coefficient expressive of the proportion of the residual length oftke bay by which the points of inflection are to be assumed shifted apart in figuring P.’for thePerry correction factor, has been determined empiricaHy, and in fihis case the best figure to uscwas found to be 0.2 in place of the 0.8 determined for the single bay. lt is natural that n shouldbe reduced, since the moments at the ends of the single bay are independent of the compression,while in a two-bay sp%r the increase of ~~Bwith increasing compression tends to shift the pointsof inflection cIoser together and so to neutralize in part the direct effect of the column effectwithin the bay in relocating those points.

The table below incIudes the actuaI correction factors, reproduced from the previous table,the factors czdculated by the Perry formula, and the ratio of the two.

Correction factors are to be applied to the maximum moments within the btiy in whic~lcolumn effect is greatest when a in that bay is 60°.

APPROXIMATIONS FOR COLUMN EFFECT IN WING SPARS 499

I-------------- Inner ___ bn------------- olIteX-...7III . . . . . . . . . . . . hulex.-~I\”. ..- . . ..--.. IuIler. -...,Y . . . . . . . . . . ...! Outer -----lW- . . . ..--... I klw._+

x- . . ..- . . . ..-. ~.....::[

XI. . . . . . . . . . . . fnner-..-.~I

L41L 36L 61L55L26L 40L 32L2iL34

LML05.96.95.S9

L 03.E3

L%l.W

—

.

1t w-U be observed that the true and approximate correction factors ne-rer difler by morethan 6 per cent unless the -nJ.ues of CYin the inner and outer bays dMer from each other by lessthan 8 per cent. If a is the same in the two b~ys, however, the approximation goes badly astr~y.To show where this critical region of equal a Iies the curve in Figure 13 has been drawn, showing

.Z Oufer bay

FIG. 13-Proportions of truss for eqnN a in two &ys

the relationship &hat must exist among the three parts of Lhe length of the spar in order that LXmay be exactIy the same in the two bays.

AH of the conclusions so far dram, and all of the discussion except th%t on the deflectioncur-res of Figure 1!27have related directly to a single particular vaIue of a. In Case 1, for ~hich

30 II IHIII III MJ ~,g 1111111 lllllt’n;e!ay:lllm uk OJ”0.)~ I I I I I I I I I 1 I I +++4/4E s~ >2.2 Iltllllllllltll<iQuu -,. 111111 ”111111

>-s :Illllltlllll LA’ I

boy

z s [.8 --+. k

Immlltwnli

$ -F

g (,4 II II IHTTTLW’ H

E 0.~ 5 111 HnlJA&vl II.$ $ II_. L-k 1 1 I I Oufer b~y

% ,5 f.o

s 8 Iltllllllllllwi506 11111 Ill [tlll~ do-< II II 111111111 \lk o= 1“1 II 111111111c1

;0 11111 Ill 11 {111 !o“ /0” 20” 300 40” 50” 60” 700 80“ 90

o!FIG.14

the deflections were computed and p~otted, the maximum moments within both bays werefigured for a number of dit’ierent -dues of that function, and a few extra moments were figured,too, for Cases H and IV. The results are tabulated below, and the ratios of &hemoments of thecorresponding figures with no compression in the spar are plotted in Figure 14.

500 REPORT NATIONAL ADVISORY COMMITTIIE FOR AERONAUTICS

MOMENTS

Inner bay 1Outer bay i Outer bay Inner bay I

a _._-l I

Case I Case I ~ Case II , Case N“

e

I

~ 001%............. 75 191 104

~ j~::::::::::::: 66 .-.. _..iij-- ,-.-..--iiF.-: 67 I

45-- . . . . . . . . . . . 67 .---.------- ------------60- . . . . . . . . . . . . 1% 65 273 154 :75-.. .-.- . . . ..- 146 55 ------------ -_---._---.-80------------- 177 42 432 231

~ The outer bay having the smaller value of a, these me the out-er-bay moments for the condition in which = in tbe inner bay is asi~dicatecl in the left-hand column. In the outer bay it is actuallyabout 10 per cent smfdler.

Although there is some crossing back and forth among the curves iu Figure 14, all are of thesame general type (except, of course, the o~e which is drawn for a noncritical bay and in whichthe correction factors drop off below unity). The effect of the cha~ge in sign of the deflectionin the outer bay in Case I is cIearIy shown. The column effect is at first positive and almostexactly large enough to balance the effect of the shift of the whole moment curve by virtue of

& .FIG.15

the increasing absolute values of the moment at the inner strut, so that the maximum momentin the bay remains substantially constant over a wide range of vaIues of a. Thenj as a isfurther increased, the spar is forced down to a negative deflection as shown in Figure 12, thecoIumn e.fleet becomes negative, and the moment in the bay drops off rapidly.

The application of the Perry formula may of course be extended irr this case over Lhe wholerange of -raIues of a, again taking n, the coefficient of shift of the points of inflection as O.2.That has been done in Figure 15, whence it wiJJbe seen that in general, as might be expected,the errors arising from the use of an approximation grow larger and larger as the compressiveload in the spar comes progressively to the “Euler lo ad.” The error is sometimes in one direc-tion, sometimes in the other, and it becomes dangerously large when a exceeds 70° or 75°.

When a exceeds 65° in either bay, or when it has very nearly the same value in the two baysand that value isin excess of 45°, it is advisabIe that a direct calculation be made for the mtixi-mum moment in that bay in which a is largest, or in both if the two vtilues are equal. Theprocess of calculation is much facilitated by the use of charts giving .3f~, as may best be shownby an example.

The inner bay in Case VII wiIl be selected for purposes of illustration, and the work carriedthrough with a equal to 60°, the figure already covered in the tabulation. The length of theinner bay in that case was 35 per cent of the total, t4at of the overhang 20 per cent, and the

APPRO~lATIONS FOR COLUIWN EFFECT IN _TG SPARS 501

bending moment ak the middle supporfi withoub cohmn effect is found by the use of the three-moment equation or from the curves in Report NTO.214 to be 152. This figure is, of course,based on a unit Ioading and a total Iength of 100 inches. Figure 11 indicates a correction factorof 1.13 to be applied to the moment when a is 60° and t-he Iength of the outer bay is 45 per cent,giving an approximate moment for 3= of 171. (R happens that the exact value of the momentwith coIumn effect has already been determined for this particular case, but the caIcuIat.ionWN be carried fihrough on the same basis as if that had not been done.)

.4 tabulation can now be made as for a representative singIe-bay spar, but in simplified form.

(A)

@)

1 ()p

—2P

=@ ----------------------------------

.34. – M.

2----------- _----- ——_—_— ——-—____ -—---

(XBN’)CSc a_________________________–_–__

i x~+lfc——Fz 2

---- ------------- —--- —------— ---

(i MB+-lf=—~— )——sec a -------------------------P ‘2

tan ,LLZ()

(A)== ------------------------------

SECJlx (= J~________________________

J&a=.( )

=(B) see ~x–~2 _____________________

60° (1.047 rad.)2.0001.1550

171

279

85.5

9S. 6

L93. 5

387

0.255

1.0320

U20

working to a degree of accuracy of individual Qures well within the scope of ai ordinaryslide-rule, th~ maxim~m moment can be cahdated with an error of not more than 1 or 2 per cenkin onIy three or four minutes more time bhan would be needed for the determination of the dis-tance between the points of inflection and the appIicat.ion of Perry’s formtia or some otherapproximation of similar nature.

The only assumption iuvo~ved in the -work just done lay in the use of Figure 11 to find MB.If hhe correction factor there plotted had been 5 per cent in error, however, so that lf~ had beenLaken as 165 instead of 173, it wouId have changed the mm%mm calculated moment in thebay only from 120 to 126, introducing an error of about the same magnitude as that made in

REST STRUT LOCXTION

lt was shown in ReporL NTO.214, on the assumption that the Perry formula could be usedto take care of the cohmm correction in both bays, that the determining condition for matiumeconomy of material in an untapered two-bay spar is that the struts shall be so pIaced that thetotal resultant stresses at the inner strut, at the outer strut, and at the point of maximummoment in the inner bay are the same. That report also includes calculations of tihe best strutIocations in terms of the relation between span, gap, and depth of spar, indicating that the besbIength of inner bay ranges from 32 to 40 per cent of the total Iength of spar, with overhangscorrespondingly ~arying between 21 and 25 per cent. Figure 13 shows that an inner bay Iengthof 32 per cent, combined with a 21 per cent overhang, makes a very slightly larger in the innerbay than in the outer, and an-y lengthening of the inner bay or shortening of the outer would,of course, increase the superiority of the inner bay m The proportion recommended iu theprevious report would therefore make the inner bay the criticaI one for coIumn effect in alIcases ~ the light of the present analysis.

The Perry formula as used in the earIier work has here been shown to give approtiatelycorrect rewdts, but no allowance was there made for the increase of WE with coIumn effect.

.

50.2 REPORT NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

Aside from the absence of such allowance, the conclusions reached in Report No. 214 aboutstrut distribution will stiH hold, as the inner bay proved the critical one by both treatments,the approximate and the exact. If the change of MB be taken roughly into account, however,it appears advantageous to increase the length of overhand about 1 per cent whiIe leaving theposition o.f the inner strut unchanged. The curves of strut distribution in Report No.214 mighttherefore wel~ be modified by increasing the figure at t ached to eact~ <UIve of equal overhand byone unit, making the best arrangement, for a truss of average aspti I ratio and gap-chord ratio,an inner bay of 34 per cent, an outer bay of 43, and an overhand of 23. These figures are givenfor a value of a of 60° in the inner bay, which is about the mean for two-bay biplanes. Anyincrease in that figure makes a still further increase of overhang advisable.

APPLICATION TO SPARS FIXED AT ENDS

Ii would be expected that the fixing moment at the inner end of a spar rigidly fixed tocenter section or fuselage wouId vary with column effect in the same general way as the momentin the inner bay, for if an increase of compression increases the upward defle.c.tion in the innerbay it must increase the curvature of the center line o.f the spar, and so the bending moments,both in the middIe of that bay and at its inner end. It would also be anticipated that theco] umn effect on the magnitudes of mid-bay moments would be less when the end of the sparis fixed than when it is pinned and free to change its slope. The single illustrative calculationmade, bearing on a spar of the same proportions as that in Case VII but with a fixed end, hasverified that expectation, for when m is 60° in the inner bay and bending moments vary in themanner shown by the tabulation below.

I~ Innermii pinned

IInner end ti~ed

t

I

Moments

~oe~~tmn ~=coo : ~tio

IX Oec&tmn ~=coo Ratio

Inner strut. . . . .r 153 ‘J

L 13 134 145 ~ 1.08Inner end . . . . . . . . . . . . . ..&~-- ..-. -iig.. ~.. -.iiG. - 8s,

! Irmer bay . . . . . . ~ ,[ __ ::! _ ::;

It is impossible to arrive at any very definite conclusion in the absence. of more computa-tions, but it seems probabIe that column effect on the fixing moment at the end of the spar willbe of little importance in two-bay spars of usual dimensions, while the correction factor to beapplied to that moment in a single-bay spar would always be less than the corresponding factorfor the mid-bay moment in the same spar. It is probable, too, that the correction factor forthe moment in the inner bay of a two-bay spar fixed at the end wiII be less than unity exceptwhen the vaIue of a in the inner bay exceeds that in the outer by at least 20 per cent, and thatin any case, whatever the ratio between those values may be, the use of the Perry formula with-out any shift of the points of inflection calculated under lateral load alone (that is, with thecoefficient n taken equal to zero) will be a safe procedure,

REFERENCES

No. 1. NE-WELL, J. S. “The Investigation of Structural lMembers under Combined Axialand Transverse Loads, Sec. I.” Air Service Information Circular No. 493. Dec. 1, 1924.

No. 2. BERRY, ARTHUR. “Calculation of Stresses in Aeroplane Wing Spars.” Trans-actions of the Royal Aeronautical Society No. 1.

PIPPARD, A. J. S. and PRITCHARD, J. L. “Aeroplane Structures.” Lo&mans,Green and Co. 1919. Pp. 116–120. (Abbreviated form of proof.) (Modified form depend-ent on different coordinates given in Reference 1.) (A similar method was developed in Ger-many during the war of 1914–1918.)

No. 3. WARNER, EDWARD P. “Wing Spar Stresses and Stress Charts.” National Advis-ory Committee for .4eronautics Technical Report hTo. 214. 1925.

No. 4. MORLEY, A. “Strength of Materials.”No. 5. WARNER, EDWARD P. “ hTotes on Wing Spar Design.” Aviation, May 20, 1922.